Ten days in Hungary: Budapest, and the two towns that explain it

Begin at Vörösmarty tér and walk south with the whole of Buda opposite. There is a Roman fort under the pavement halfway along that nobody stops for.

The first permanent bridge over the Danube here, opened in 1849, designed by an English engineer who built a smaller version over the Thames at Marlow. Walk to the middle and back.

Sixty pairs of iron shoes cast onto the stone edge of the river, where Arrow Cross militiamen made Jewish Budapesters take them off before shooting them into the water in the winter of 1944 and 1945. Five minutes north of the bridge.

The fifth and seventh districts both work. Eat late; Hungarians do, and the kitchens stay open.

Seven white turrets along the eastern wall, one for each of the Magyar tribes, finished around 1902 as a viewing terrace and pretending hard to be medieval. The lower levels are free and the view from them is the same one you have seen on the postcards.

Late Gothic, heavily restored in the nineteenth century, with every interior surface covered in painted pattern and a tiled roof that is the best thing on the hill from outside. Hungarian kings were crowned here. Cover your shoulders.

Walk two blocks west of the church, away from the bastion, and the prices halve without the quality following them down.

The palace wing at the far end of the ridge holds the national collection, and the case for spending an afternoon in it is the nineteenth century Hungarian painting, which almost nobody outside the country has seen. The dome gives you the second best view on the hill.

Neo-Gothic, 270 metres long, finished in 1902, and still the largest building in Hungary. The coronation regalia are inside and the staircase does not photograph. Guided tours only.

A Neoclassical block with the country's story from the Romans to 1990 laid out inside, and the steps where the 1848 revolution was proclaimed. Worth ninety minutes here for the sense it makes of everything you will see in Pécs and Eger later.

The street food and the small kitchens in the old Jewish quarter are the best value eating in central Budapest.

The largest synagogue in Europe, built in the 1850s to a Moorish design by a Viennese architect, with two onion domes 43 metres high. The ticket includes the museum and the memorial garden behind, which is a cemetery it should never have needed to be.

The colonnade of Hungarian kings and statesmen at the head of Andrássy út, put up for the thousandth anniversary of the Magyar arrival in 1896. Read the plaques: half the country's arguments about itself are settled on this square in bronze.

A castle that is not one: a full scale stage set built for the 1896 exhibition, with Gothic, Romanesque and Baroque wings copied from real buildings around the country and arranged around a moat so you only ever see one at a time.

Two full restaurants and a row of kiosks inside the park. Keep it light before the baths.

The huge yellow bath complex at the top of the park: three outdoor pools and a dozen more inside, saunas and steam rooms in the ticket, and the hot outdoor pool with the chess players that everyone has seen a photograph of.

From Déli or Keleti, about two hours forty on a fast service. Sit on the right hand side once the line leaves the plain.

Built in the sixteenth century by Pasha Qasim from the stones of the Gothic church that stood here first, and Catholic again since the Ottomans left. Inside, under a bare dome, the mihrab and painted Qur'anic verses survive next to the altar.

A pharmacist who heard a voice at twenty seven telling him he would surpass Raphael, gave up the shop, walked around the Mediterranean and the Middle East and came back with enormous luminous canvases. Hungary's strangest painter, almost all of him in this one house.

Terraces on the upper half of Széchenyi tér face the dome, and a student city keeps them cheap.

Fourth century Christian burial chambers from Roman Sopianae, painted with Adam and Eve and the Virgin, and the largest surviving complex of its kind in the old European provinces. Seven are open under the modern visitor centre. UNESCO listed since 2000.

Four corner towers over an eleventh century core, altered by Gothic, Renaissance and Ottoman hands and then scraped back to an imagined Romanesque in the 1880s. The crypt is the oldest and the best of it.

The self service places the students use serve a full hot lunch for the price of a Budapest coffee, and they are genuinely good.

Vilmos Zsolnay's factory invented eosin, an iridescent glaze that shifts from green to gold as you move past it, and won a Grand Prize in Paris in 1878 with it. After a room of this you will spot the roof tiles on half the good buildings in Hungary.

The old works, reopened when Pécs was a European Capital of Culture in 2010. Prioritise the family and factory history rooms and the mausoleum; the interactive science house is there for anyone travelling with children.

Buses run back to the centre every few minutes, and a taxi across town costs less than a station sandwich.

Victor Vasarely was born in this city and effectively invented op art. One of only three collections of his work anywhere, and an hour of it is either delightful or a headache, with very little in between.

Roughly five to six hours door to door. Hungarian rail radiates from the capital, so there is no way round it.

Around two hundred cellars cut into soft volcanic rock around one horseshoe valley, most of them family holdings, many pouring straight from the barrel by the decilitre. Do not drive; a taxi back up the hill costs very little and the walk is half an hour, steeper than it looked coming down.

Several of the larger cellars serve full meals, which is the sensible way to handle an evening of tasting after a day on trains.

The exhibition in the Bishop's Palace tells the siege and what came after it properly, but the casemates cut through the rock underneath are the better half of the visit. The ramparts hand you the whole town in one look.

The square under the castle, and not a trap: game and river fish are the local strengths and the weekday set menus are a third of evening prices.

A tapering brick column just over forty metres high, left standing in a Baroque street after the mosque at its foot came down, and the northernmost medieval minaret in Europe. Ninety seven steps up an extremely tight spiral, so admire it from the pavement if enclosed spaces are not for you.

The Baroque college opposite the basilica, for two rooms: the archbishop's library, with a ceiling fresco of the Council of Trent and the only handwritten Mozart letter in Hungary, and the observatory tower, where a camera obscura throws a live picture of the streets onto a table in a dark room.

Built between 1831 and 1836 by József Hild, who went on to design the basilica in Budapest. Free to walk into, and there are organ recitals most mornings in season at half past eleven.

Direct services take a little over two hours; the intercity via Füzesabony is quicker and needs one change. The airport is a bus ride from the centre at the far end.